I also had never seen a spay performed like that before until I moved to Scotland. Apparently it is a very popular technique here and over 90% of vets tend to perform their spays that way in the UK. I think it all comes down to what technique a vet is taught in school and what they are comfortable with.
Historically, spays were performed on the flank to ensure better wound healing and thus reduced risk of wound breakdown, but with the advent of new suture materials more appropriate for the linea alba, it kind of negates this risk. Personally if I had a choice, I would opt for the midline approach because you get a better visualization of all the structures within the abdomen and decrease the risk of leaving ovarian remnants behind, or ligating the ureter instead. Midline incisions are also thought to be less painful because you are going through the midline (linea alba) which has less pain receptors as opposed to going through 3 layers of muscle via the flank approach. The pros to doing a flank spay include being able to monitor the wound from afar in fractious cats and perhaps decreased tension on the wound site/risk of evisceration, since there is not as much pressure on the area (i.e. all the abdominal contents aren’t pressing down on the wound).
All in all, it really just comes down to what someone is comfortable doing. Now, clipping up the cat's front leg with scissors though is just weird! I have definitely never see that before and it made me squirm to watch it!